NO. 8.] REMARKS ON THE EARTH'S CRUST. 75 



terms, on the other hand, vary with 9; for a certain value of the latter, their 

 sum equals 0. For smaller values their sum is negative, and its numerical 

 value becomes greater and greater the nearer 6 approaches 0, in which case 

 the numerical value becomes equal to the first term. For greater values of 

 6, the sum becomes positive, and increases continually with 9, until for 9 = n 

 it becomes equal to the first term. An idea of the value of 6 for which the 

 sum of the two terms equals may be obtained by the aid of the following 

 proposition, which is easily proved. If a point P is situated outside a spherical 

 shell with constant density and its centre at 0, a spherical surface with the 

 line OP as its diameter will divide the shell into two parts, which will each 

 exert an equal influence upon P, and thus an influence equal to half that 

 of the entire shell. Thus the thinner the shell, the smaller will be the angular 

 radius, 9, of the conical section, which will exert an influence equal to half 

 that of the entire shell upon a point in the centre of the limiting zone. 



By the aid of the above expression, we will try to determine the influence 

 of a similar conical section of the above-mentioned continental added masses 

 upon a point on the earth's surface in the centre of the limiting zone. We will, 

 as before, assume the thickness of the earth's crust to be fej = 0'02 .R , and 



the depth of the sea, fe 2 ==r 3'5 km. = .-,... R . * For the sake of simplicity 



we will assume the densities, ((>, 1) and (p, Q), to be constant in both 

 layers. This will make no perceptible change, as the numerical value of 

 fei -~ P)' according to what we have said above with regard to the con- 

 stitution of the ocean-bed, is probably smallest not only deepest down in the 

 base, but also above just below the covering. The following diagram 

 represents graphically the result of the calculation. The angular radius of the 

 zone is the abscissa, and the ordinate gives the alteration in the acceleration of 

 gravity dependent upon the section. Each division indicates 0'02 mm. It 

 will be seen that even for an angular radius of 20, which will correspond 

 to a radius of about 2200 km., the interior negative masses will not altogether 

 succeed in neutralising the attraction of the external positive shell. The aggre- 

 gate effect of the section, however, is only about 0'07 mm. 



1 The numerical values here chosen are of no importance for the following investigation, 

 only supposing that the thickness of the crust is considerable in relation to the depth 

 of the ocean, and very small in comparision to the earth's radius. 



